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Journal of Mathematical Finance, 2017, 7, 219-237
http://www.scirp.org/journal/jmf
ISSN Online: 2162-2442
ISSN Print: 2162-2434
DOI: 10.4236/jmf.2017.71012 February 28, 2017
Partially Adaptive and Robust Estimation of
Asset Models: Accommodating Skewness and
Kurtosis in Returns
James B. McDonald1, Richard A. Michelfelder2
1Brigham Young University, Provo, USA
2Rutgers University, Camden, USA
Abstract
Robust regression estimation deals with selecting estimators that have desir
a-
ble statistical properties when applied to data drawn from a wide range of di
s-
tributional characteristics.
Ideally, robust estimators are insensitive to small
departures from the assumed
distributions and hopefully would be unbiased
and have variances close to estimators based on the true distribution. The a
p-
proach explored in this paper is to select an estimator based on a flexible di
s-
tribution which includes, for example, the normal as a limiting case.
Thus, the
corresponding estimator can accommodate normally distributed data as well
as data having significant skewness and kurtosis.
In the case when an assumed
distribution over-parameterizes the true distribution, the variance of the est
i-
mator is larger than necessary, but often the increases are modest and much
smaller than assuming a model which does not include the true distribution.
The selection of a flexible probability distribution can impact the efficiency
and biasedness of the corresponding robust estimator.
Knowing the relations
among potential distributions can lead to a better estimator that improves e
f-
ficiency, avoids bias, and reduces the impact of misspecification.
Keywords
EGB, SGT, IHS, g-and-h, Flexible Densities, Robust Estimation
1. Introduction and Background
The problem of selecting an appropriate probability distribution for a dataset, if
one exists, is as old as Gauss’s development of the “normal” distribution. Many
distributions have been developed since the normal was used to describe the dis-
tribution of a sample drawn from an unknown population. Important differenc-
How to cite this paper:
McDonald, J.B.
and
Michelfelder, R.A. (2017)
Partially
Adaptive and Robust Estimation of Asset
Models: Accommodating Skewness and
Kurtosis in Returns
.
Journal of Mathema
t-
ical Finance
,
7
, 219-237.
https://doi.org/10.4236/jmf.2017.71012
Received:
December 23, 2016
Accepted:
February 25, 2017
Published:
February 28, 2017
Copyright © 201
7 by authors and
Scientific
Research Publishing Inc.
This work is licensed under the Creative
Commons Attribution International
License (CC BY
4.0).
http://creativecommons.org/licenses/by/4.0/
Open Access
J. B. McDonald, R. A. Michelfelder
220
es between many distributions are the implied restrictions on the higher order
moments, such as skewness and kurtosis.
The modern literature of the empirical analysis of fitting alternative probabil-
ity distributions to financial data begins with Mandelbrot [1] and Fama [2] and
continuing to this day concludes that distributions of financial asset prices or the
corresponding rate of return (returns) data are usually asymmetric and are short
and wide compared to the normal pdf. That is, they are skewed and leptokurtic.
Therefore, methods that are used to estimate the relationship between assets
prices and returns such as ordinary least squares (OLS) regression or those that
are based on an assumed symmetric and non-leptokurtic normal pdf may gen-
erate inefficient parameter estimates when the errors are not normally distri-
buted. The inappropriate assumption of a symmetric distribution for the error
can lead to biased estimates of the intercept if the data are skewed. There is a
growing literature on robust estimation and outlier resistant methods that deals
with how to address the potential inefficiency and bias problems associated with
outliers and non-normality often found in financial data. One approach is to as-
sume a more general pdf that can accommodate asymmetry and thick tails.
Many such pdf’s and outlier resistant methods can be nested using a family of
flexible pdf’s whose members are obtained by imposing different restrictions on
the distributional parameters with corresponding restrictions on the moments.
Some of these generalized pdf’s can be visualized as being at the top of a pyramid
of pdf’s obtained by imposing parameter restrictions which imply different restric-
tions on feasible values of skewness and kurtosis. If a model is over-parameterized
(selecting a more general model than necessary), then the estimators will be in-
efficient. For example, if the data are normally distributed and a five-parameter
Skewed Generalized t (SGT) distribution is fit to the data where only two para-
meters are needed to describe the data, the estimators of the SGT parameters will
not be efficient. However, if a symmetric distribution is fit to skewed data, the
parameter estimates and implied estimated moments will be biased. By studying
the families of flexible distributions and their related characteristics, the re-
searcher will be better able to select a robust estimator having desirable statistical
properties. For example, McDonald, Michelfelder, and Theodossiou [3] showed
that if OLS or a robust estimator based on a symmetric error distribution is used
to estimate the relationship between asset prices and returns, the estimated in-
tercept will be biased if the errors have a skewed distribution.
This paper presents the relationships between the generalized and the re-
stricted pdf’s associated with several families of pdf’s that are more commonly
being used in empirical finance. We also report the feasible skewness-kurtosis
spaces of the generalized pdf’s and compare them with empirical estimates of US
stock returns. These results add to the financial estimation literature by showing
the nesting relationships within the flexible pdf’s and the corresponding restric-
tions on higher order moments. It also demonstrates the performance of the ge-
neralized pdf’s in fitting non-normal datasets. It is important to note that the
feasible skewness-kurtosis space corresponding to a generalized pdf does not
J. B. McDonald, R. A. Michelfelder
221
accommodate all possible skewness-kurtosis combinations.
2. Literature Review
Mandlebrot [1] and Fama [2] initially found that stock returns regression resi-
duals are skewed and are fat-tail distributed. McDonald and Nelson [4] found
that many stock risk premiums are positive or negatively skewed and most have
thick tails. Harvey and Siddique [5], Harvey and Siddique [6], and Chan and
Lakonishok [7] concluded that stock returns are skewed and fat-tail distributed
and applied various robust methods to address the estimation inefficiencies.
Chan and Lakonishok [7], Butler, McDonald, Nelson, White [8], McDonald
and Nelson [4] discuss specifically the inefficiency in estimating the CAPM beta
with OLS.
There are many robust estimation methods, some that are outlier adjustment
methods, others are based on alternative specifications of pdf’s and some that are
both, such as least absolute errors (using the Laplace pdf) rather than least
squares for regression estimation. This investigation focuses on those methods
that use alternative pdf’s that can accommodate varying levels of skewness and
kurtosis and that nest more restrictive pdf’s.
Boyer, McDonald, and Newey [9] bifurcate robust estimation into reweighted
least squares or least median squares, and partially adaptive estimators. The ro-
bust or partially adaptive estimators considered in this paper can be viewed as
quasi-maximum likelihood estimators. They maximize a likelihood function
corresponding to an approximating error distribution to yield estimated regres-
sion and distributional parameters. The least squares methods address only the
choice of regression parameters. Boyer, McDonald, and Newey [9] use simula-
tions to compare the efficiency of generalized pdf’s and least squares methods.
Using one of the generalized pdf’s, they concluded that generalized pdf’s pro-
duced more efficient estimators than outlier adjustment methods that cannot
change pdf parameters when regression errors have skewness or kurtosis.
Therefore, among the myriad of robust estimation methods, this paper focuses
on the use of generalized pdf’s.
The generalized probability distribution families considered in this investiga-
tion can accommodate a wide range of data characteristics. These generalized
probability distribution families are the generalized beta and exponential genera-
lized beta and variants from McDonald and Xu [10], the skewed generalized T
from Theodossiou [11], the inverse hyperbolic sine from Johnson [12] and the
g
-and-
h
from Tukey [13] and Dutta and Babel [14]. Some of these distributions
have been used in Hansen, McDonald, and Theodossiou [15] to model various
skewed and fat-tail distributed financial time series data in GARCH specifica-
tions. The skewed generalized T pdf is starting to be used more frequently and
recently has been as added to the Stata© econometric software package for re-
gression. The SGT was used by Hansen, McDonald, Theodossiou, and Larsen
[16] to show the differences in regression results for a model of real estate prices
with data and errors that are positively skewed and fat tail distributed. It clearly
J. B. McDonald, R. A. Michelfelder
222
shows the improvement in variance of the estimates.
Mauler and McDonald [17] apply the generalized beta of the second kind, the
inverse hyperbolic sine, the
g-
and-
h
, and others to generalize the Black-Scholes
option pricing model to explore potential improvements relative to the original
log-normal specification of the options model. All alternative flexible pdf’s con-
sidered generated improvements in the accuracy of options price estimates rela-
tive to the log-normal pdf.
Kerman and McDonald [18] [19] derive feasible skewness-kurtosis spaces for
variants of pdf’s within the exponential generalized beta and the skewed genera-
lized T families. Kerman and McDonald [19] find that the skewed generalized T
and its nested skewed generalized error pdf’s have the most flexibility of many
pdf’s that they modeled. McDonald, Sorenson, and Turley [20] obtain expres-
sions defining the skewness-kurtosis spaces corresponding to the generalized
beta of the second kind.
Theodossiou [21] derives the skewed generalized error distribution nested
within the skewed generalized T family and applies it to various asset pricing
models estimations and derivations. Theodossiou and Savva [22] use robust es-
timation (partially adaptive estimators) based on the skewed generalized T,
which accommodates negatively skewed asset returns, to address empirical in-
consistencies in the finance literature on the risk-return relation. Next, the fami-
lies of the generalized pdf’s are discussed.
3. Families of Generalized Probability Distributions for
Financial Modeling
We present the following generalized pdf families that accommodate asymmetry
and thick tails and the pdf’s that they nest where
y
is the random variable and
the distributional parameters control the moments of the distribution. Whereas
the normal has two parameters, the following distributions have 4 to 5 parame-
ters (described in the Appendix for each distribution):
1) The generalized beta (GB),
( )
GB ; , , , ,yabc pq
,
2) The exponential generalized beta (EGB),
( )
EGB ; , , , ,ym cpq
φ
,
3) The skewed generalized T (SGT),
( )
SGT ; , , , ,ym pq
λφ
,
4) The inverse hyperbolic sine (IHS),
( )
IHS ; , , ,yk
µσ λ
and,
5) The
g-
and-
h
distribution,
( )
GH ; , , ,yabgh
.
The Appendix shows the specifications of the pdf’s and the associated para-
meter expressions that controls their shape (skewness and kurtosis) and loca-
tion. The GB, EGB, and SGT are five-parameter distributions and the IHS and
g-
and-
h
distributions each involve four parameters. All of these families nest the
normal or a variant of the normal. For example, the GB nests the half-normal.
Gauss’s development seems to have been the catalyst which motivated future
generations of mathematicians and statisticians to start with the normal pdf and
generalize it, going down different pathways, to better model the diverse distri-
butional characteristics encountered in modelling various data sets.
Figures 1-5 show the many distributions that are nested within the five pdf’s
J. B. McDonald, R. A. Michelfelder
223
Where:
a controls peakedness
;
b is a scale parameter
;
c domain
( )
01
aa
yb c
<< −
;
p
,
q shape parameters.
Figure 1. Generalized beta family of density functions [10].
Where:
m
controls location;
φ
is a scale parameters;
c
defines the domain;
p
,
q
are shape parameters.
Figure 2. Exponential generalized beta family of density functions [10].
J. B. McDonald, R. A. Michelfelder
224
Where:
m
= mode (location parameter);
scale
φ
=
;
1
skewness area to left of , 1 1
2
m
λ
λλ
−
= = −< <
;
p
,
q
= shape pa-
rameters (tail thickness, moments of order <
pq
= d
f
).
Figure 3. Skewed generalized T family of density functions [3].
Parameter Values
Shape of Distribution
μ
(mean) Location
σ
2
(variance) Dispersion
λ
(skewness) ≠ 0
Asymmetry
k
(tail thickness) Thick Tails
λ =
0
and
k → ∞
Approximates Normal
Figure 4. Inverse hyperbolic sine family of density functions.
Parameter Values
Shape of Distribution
g
≠ 0 Asymmetry only
h
≠ 0 Thick tails only
g =
0
and
h =
0 Approximates Normal
g
≠ 0
and
h
≠ 0 Asymmetry and Thick Tails
Figure 5. The
g
-and-
h
family of density functions.
listed above. An inspection of Figures 1-5 show that the nested distributions are
obtained by imposing various restrictions on the parameter values of the more
flexible pdf. For example, the restrictions on the values of
p
and
q
of the GB and
EGB control the skewness and kurtosis of those families of pdf’s. For example, if
c
= 1 in the GB distribution (Figure 1), the corresponding pdf is seen to be the
generalized beta of the second kind (GB2) which is defined for positive valued
random variables. For the SGT family in Figure 3,
λ
describes skewness (nega-
tive for negative skewness and vice versa) and
p
and
q
determine the shape of the
pdf. The IHS and
g
-and-
h
pdf’s nest the normal (see Figure 4 and Figure 5) as
limiting cases and have the flexibility to accommodate a wide range of skewness
and kurtosis values.
The GB nests at least 26 pdf’s. Among those some that are commonly used in
J. B. McDonald, R. A. Michelfelder
225
economics and finance are the GB2, log-normal (LN), Pareto, truncated or half-
student’s T, chi-squared, exponential (EXP) and the truncated or half-normal
pdf’s. Its exponential version, the EBG, nests the EGB2, which has been used in
recent papers on robust estimation involving the capital asset pricing model and
other financial models. The EGB and SGT nest at least ten distributions each.
The SGT, among the many others, nests the Laplace, uniform, normal, student’s
T, and the generalized error distribution (GED). The non-unitary version of the
student’s T and the GED are offered as options to the normal pdf by EVIEWS©
and Stata© econometric software for GARCH regression error pdf choices to
model thick-tailed errors distributions. As this writing, no pre-written commer-
cially available statistical software is available for estimation that we are aware of
for most of these generalized pdf’s other than the SGT in a regression specifica-
tion in Stata© (sgtreg).
The combination of mathematically admissible skewness-kurtosis values cor-
responding to the generalized pdf’s, the EGB2, SGT, SGED, IHS and
g
-and-
h
pdf’s are shown on Figure 6. The
g
-and-
h
has the least restrictive combination
of the admissible moments and the EGB2 is the most restrictive with all combi-
nations having to be on or inside the EGB2 moment space “smile.” differences in
minimum levels of kurtosis.
Figures 7-12 show how the shapes of the density functions change for varying
values of the parameters of each pdf. Note in Figure 12 that for the
g
-and-
h
we
allowed
h <
0 which corresponds to a random variable with bounded support
and permits bimodal distributions. Combined with varying skewness values for
g
, the pdf’s have bounded support, but only for
g <
0
.
4. Empirical Applications
4.1. Distributional Characteristics
First, we consider the distributional characteristics of the total stock returns for
the population of stocks included in the Center for Research for Security Prices
(CRSP) database. Secondly, we focus on the distribution of the stock returns on
two stocks, one normally distributed and the other non-normally distributed.
We also look at the impact of the distribution on estimated capital asset pricing
model betas. Finally, we consider the distributional impacts in an ARCH speci-
fication.
The data used is the daily, weekly, and monthly excess stock returns for all
continuously traded common stocks in the CRSP database for every trading day
for five years within the period January 2, 2002 to December 29, 2006 with ap-
proximately 1250 daily returns, 260 weekly returns and 60 monthly returns data
points for each stock. Since asset market speculative bubbles and crashes have a
tendency to exacerbate skewness and thick tails of the returns distribution, this
investigation chose a time frame that did not include either asset market condi-
tions. The financial market crisis and the ensuing extreme drop in asset prices
that occurred in the forthcoming years after 2006 were purposely avoided so that
returns in a typical market regime are modeled. The choice of an observation
J. B. McDonald, R. A. Michelfelder
226
period that includes bubbles would exacerbate the difference in results between
robust and standard estimation methods.
Figure 6. Skewness and kurtosis ranges for the EGB2, SGT, SGED, IHS and
g
-and-
h
distributions.
Where: a controls peakedness;
b
is a scale parameter and
p q
are shape parameters.
Figure 7. GB2 pdf’s evaluated for different parameter values.
J. B. McDonald, R. A. Michelfelder
227
Where:
m
controls location;
φ
is a scale parameters;
p
,
q
are shape parameters.
Figure 8. EGB2 pdf’s evaluated for different parameter values.
Where:
m
= mode (location parameter);
scale
φ
=
;
1
skewness area to left of , 1 1
2
m
λ
λλ
−
= = −< <
;
p
,
q
= shape parameters (tail thickness,
moments of order <
pq
= d
f
).
Figure 9. SGT pdf’s evaluated for different parameter values.
J. B. McDonald, R. A. Michelfelder
228
Where:
2
mean variance skewness parameter tail thickness.k
µσ λ
= = = =； ； ；
Figure 10. IHS pdf’s evaluated for different parameter values.
Figure 11.
g-
and-
h
pdf’s evaluated for different parameter values with
h
> 0.
J. B. McDonald, R. A. Michelfelder
229
Figure 12.
g-
and-
h
pdf’s evaluated for different parameter values with
h
< 0.
The excess return is calculated by CRSP as the total holding period rate of re-
turn minus the total holding period rate of return on the one-month US Trea-
sury (the Fama-French risk free rate of return). This provided data for 4547
stocks traded on the NYSE, NASDAQ, and AMEX exchanges for the time frame.
The skewness and kurtosis values were calculated for the daily, weekly and
monthly returns for each stock for the time frame.
Figure 13 and Figure 14 show the plots of the estimates of the skewness and
kurtosis contrasted with the admissible parameter spaces for each pdf. It shows
how high to lower frequency returns affect the skewness and kurtosis of stock
returns. ARCH processes in returns are more pronounced for higher frequency
data therefore we should expect to see more leptokurtosis relative to skewness
from ARCH effects as intermittent high and low volatility in returns clusters
drive the persistence of the volatility of returns while the randomness of the al-
gebraic signs of the spikes (+ or −) dampens skewness in either direction.
A comparison of the admissible moment spaces with the estimated moments
shows that much of the data does fall within skewness-kurtosis feasible spaces of
the pdf’s. Figure 15 shows the proportion of the estimates that fall within the
parameter spaces for the selected generalized pdf’s. It shows that the
g-
and-
h
admissible space includes nearly all of the estimates with 100%, 99.98% and
98.99% of the daily, weekly and monthly estimates, respectively, falling within
the space. The EGB2 has the least fit of the estimates with 15.48%, 43.81% and
J. B. McDonald, R. A. Michelfelder
230
50.80% fitting within its admissible space. The SGT and IHS spaces generally fit
the estimates well with roughly 80% to 90% of the estimates within their spaces.
This finding is consistent with Kerman and McDonald (2013) that the SGT has
the most flexibility of the EGB2, SGT, and IHS families. The “bound” in these
figures corresponds to Klassen [23] bound for unimodal distributions.
Figure 13. Daily, weekly and monthly excess returns moments and admissible skewness –
kurtosis parameter spaces.
J. B. McDonald, R. A. Michelfelder
231
Figure 14. Monthly excess returns moments and admissible skewness – kurtosis parame-
ter spaces.
Figure 15. Fraction of stock returns in admissible parameter space.
4.2. Two Examples
We now contrast the pdf’s of two stocks with normally and non-normally dis-
tributed returns. Figure 16 and Figure 17 show the empirical pdf’s of the total
returns for US Steel and iShares as examples. US Steel was chosen because its
returns are approximately normally distributed with almost no skewness or
excess kurtosis as reflected in the statistically insignificant value of the Jar-
que-Bera (JB) statistic, which is asymptotically distributed as a chi-square with
two degrees of freedom. iShares was chosen because it has severe skewness
(−29.1), kurtosis (965.1), and a statistically significant JB statistic equal to
48,733,899. Note that the plotted pdf’s, log-likelihood values, sum of squared er-
rors (
SSE
) and sum of absolute errors (
SAE
) as indicators of goodness-of-fit for
all pdf’s for US Steel returns are very similar in value. This is in sharp contrast to
the results for iShares. The log-likelihood value of the flexible pdf’s are orders of
magnitude higher than that for the normal. The fitted pdf’s,
SSE’s
and
SAE’s
all
indicate that the flexible pdf’s provide a much better fit than does the normal.
daily weekly monthly
EGB2 15.48% 43.81% 50.80%
IHS 83.92% 84.39% 61.97%
SGT 87.62% 89.00% 95.10%
g-and-h 100.00% 99.98% 98.99%
J. B. McDonald, R. A. Michelfelder
232
These two examples demonstrate how much better the pdf’s that accommodate
skewness and kurtosis can approximate the distribution of the returns relative to
the normal.
4.3. Capital Asset Pricing Model Betas
We have also performed capital asset pricing model (CAPM) regressions for two
stocks, one with approximately normally distributed regression errors and the
other that is skewed and has thick-tails. This is the same approach used by
McDonald, Michelfelder, and Theodossiou (2010) to compare beta (slope) esti-
mates for public utility stocks with normal and skewed and thick-tail distributed
regression errors. Figure 18 shows the skewness, kurtosis, JB Statistics, and beta
estimates for two stocks, one with normally distributed regression errors and the
other non-normally distributed. United Natural Foods has normally distributed
CAPM regression errors as indicated by the values of skewness, kurtosis and the
JB statistic. The OLS beta for United Natural Foods is 0.313 and the range from
the other regression error pdf’s range from 0.302 to 0.335.
Figure 16. PDF fits for a stock with normally distributed daily excess returns: US steel.
Estimated PDF logL SSE SAE Chi ^2
Normal 2753.52 0.001 0.12 27.81
EGB2 2756.83 0.001 0.11 23.38
IHS 2756.76 0.001 0.11 23.46
SGT 2758.78 0.001 0.12 28.19
J. B. McDonald, R. A. Michelfelder
233
Figure 17. PDF fits for a stock with leptokurtic and skewed daily excess returns: iShares.
Figure 18. Capital asset pricing model beta estimates for stock examples with normal and non-normally distributed returns re-
gression error terms.
The 99 Cent Only stock returns distribution is non-normally distributed as
indicated by the skewness and kurtosis values and JB statistic. The beta esti-
mated with OLS for the 99 Cent Only stock is subject to more prediction error
compared with United Natural Foods as the OLS estimate is 0.184 and the range
for the flexible pdf’s are from 0.106 to 0.125.
Estimated PDF logL SSE SAE Chi ^2
Normal 2516.86 0.099 0.93 1433.33
EGB2 3713.99 0.002 0.13 43.47
IHS 3795.21 0.001 0.12 33.43
SGT 3810.07 0.003 0.21 79.35
Company Name Skewness Kurtosis JB stat
UNITED NATURA L FOODS INC -0.074 2.8004 0.1543
99 CENTS ONLY STORES 1.7541 7.6594 85.0456
Statistics of OLS residuals
Company Name OLS TGT SGED EGB2 IHS ST SGT
UNITED NATURA L FOODS INC 0.313 0.313 0.335 0.334 0.303 0.302 0.314 0.335
99 CENTS ONLY STORES 0.184 0.125 0.125 0.110 0.109 0.106 0.110 0.110
Estimated Betas
J. B. McDonald, R. A. Michelfelder
234
4.4. ARCH Specifications
Lastly, we consider the impact of distributional assumptions in an ARCH speci-
fication. Figure 19 shows the root mean square errors of the estimated beta from
10,000 replications of 60-month simulations for the three data generating
processes (DGP). The data generating process is
( )
0 0.9 excess market returns
tt
t
y
ε
=++
where
1, , 60t=
months between
1 2002
to
12 2006
. For the normal-no ARCH
()
2
~ 0,
t
N
εσ
,
for normal-
ARCH
,
( )
0.5
2
0 11
tt t
aa
εµ ε
−
= +
with
( )
0,1
t
N
µ
and for the T-ARCH
,
( )
0.5
2
0 11
tt t
aa
εµ ε
−
= +
with
( )
~5
t
T
µ
.
Not surprisingly, as the shaded highlights show for each of three data gene-
rating processes, the correct specification yields the most efficient estimates. For
example, consider the normal-no-ARCH DGP. Over-specifying the model (us-
ing a more flexible pdf than necessary) increases the variance of the estimates
(reduces efficiency). However, in many cases the efficiency loss is modest. This is
also true for the ARCH estimations for this data generating process, with addi-
tional efficiency losses associated with the inclusion of unnecessary ARCH pa-
rameters. The normal-ARCH DGP results also show that over-specifying the
model increases the root mean square error whereas correctly including the
ARCH component improves estimator efficiency. Neglecting to account for the
ARCH component has a significant impact whereas specifying a more flexible
pdf has a modest impact on the RMSE in most cases.
Regarding the T-ARCH DGP, again, as expected, the correct specification
yields the most efficient estimates. Again, failing to account for the ARCH com-
ponent has a greater negative impact on efficiency than does over parameteriz-
ing the underlying distribution.
Therefore, correctly specifying the data generating process yields the most ef-
ficient estimator as measured by RMSE. Over-specifying the error distribution,
including the inclusion of an unnecessary ARCH component reduces efficiency,
but in many cases the impact is small. Similarly, failure to include an appropriate
Figure 19. Root mean square errors based on simulations of the prediction of excess
stock returns.
Errors
Estimation Non-ARCH ARCH Non-ARCH ARCH Non-ARCH ARCH
OLS/Normal 0.352 0.356 0.347 0.291 0.353 0.300
LAD 0.444 0.446 0.397 0.369 0.315 0.297
T 0.358 0.363 0.338 0.293 0.283 0.265
GED 0.381 0.389 0.357 0.318 0.306 0.285
GT 0.387 0.396 0.362 0.322 0.306 0.286
SGED 0.406 0.417 0.374 0.341 0.318 0.297
EGB2 0.371 0.376 0.352 0.312 0.300 0.281
IHS 0.368 0.377 0.348 0.319 0.291 0.275
ST 0.375 0.382 0.350 0.310 0.293 0.277
SGT 0.409 0.420 0.376 0.344 0.316 0.297
Root Mean Square Error (RMSE) for 10,000 replications
N(0, σ^2)
N(0,1), Arch(1)
t(5), Arch(1)
J. B. McDonald, R. A. Michelfelder
235
ARCH component reduces efficiency. Log-likelihood ratios or Wald test statis-
tics can help detect over-specification of an error data generating process.
5. Conclusion
Robust or partially adaptive estimation is an approach to estimating parameters
which are relatively insensitive to mis-specifying the underlying distributional
assumptions of the model. We have shown several families of general or flexible
distributions that can reduce the impact of model misspecification. It is also
important to understand that the more general distributions, while accommo-
dating possible skewness and thick tails, cannot accommodate all possible com-
binations of skewness and kurtosis parameter values. The wrong choice of an
error distribution can reduce efficiency as well as introduce bias to the estimates.
This paper shows the family trees, nesting relations, parameter space restrictions
and a few asset returns applications of the major flexible pdf’s used in robust es-
timation in the literature. A researcher must choose very carefully the appropri-
ate distribution. The choice of a more general pdf has an increased likelihood of
including a correct specification.
Acknowledgements
We are grateful to Brad Larsen for his excellent research assistance. Brad Larsen
is currently an assistant professor of economics at Stanford University. We also
thank participants at various Multinational Financial Society Annual Confe-
rences where some of this material was first presented.
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Appendix: Specifications of the General Probability Distributions and Their Parameters
Distribution
Specification
Parameters
Generalized
Beta (GB)
()( )( )
( )
( ) ( )
( )
( )
1
111
GB ; , , , , , 0 1
,1
q
a
ap
aa
pq
a
aq
ay c yb
yabc pq y b c
b B pq c yb
−
−
+
−−
= << −
+
a controls peakedness
b is a scale parameter
c domain
( )
01
aa
yb c<< −
p
,
q shape parameters
Exponential
Generalized
Beta (EGB)
( )
( )
( )
( )
( )
( )
( )
( )
1
e 11 e
EGB ; ,, , , , , 1e
q
pym ym
pq
ym
c
ym cpq B pq c
φφ
φ
φφ
−
−−
+
−
−−
=+
m
controls location
φ
is a scale parameter
c
defines the domain
p
,
q
are shape parameters
Skewed
Generalized
T (SGT)
( )
( ) ( )
( )
( )
1
1
SGT ; , , , ,
2 ,11 sign
1
p
pp
qp
p
y m pq
p
ym
qB q ym q
p
λφ
φλφ
+
−
++−
=
m
= mode (location parameter)
scale
φ
=
1
skewness area to left of , 1 1
2
m
λ
λλ
−
= = −< <
p
,
q
= shape parameters (tail thickness, moments of
order <
pq
= d
f
)
Domain:
( )
,−∞ ∞
Inverse
Hyperbolic
Sine (IHS)
Where:
( )
( ) ( ) ()
()
()
222
22
ln ln
2
2
2 22
e
HIS , , , , 2π
kyy
k
yk y
µ δσ σ θ µ δσ σ λ θ
µσ λ
θ µ δσ σ σ
− −+ + + −+ − +
=
+ −+
( )
2
0.5
1 , , 0.5 e e e ,
k
w www
λλ
θ σ δ µσ µ
−
−
= = = −
and
( ) ( )
22 2
0.5 0.5
22
0.5 e e 2 e 1
kk k
w
λλ
σ
−− −
+ −+
= ++ −
2
mean
variance
skewness parameter
tail thicknessk
µ
σ
λ
=
=
=
=
Domain:
( )
,−∞ ∞
g-
and-
h
( )
2
2
,
e1
e
gZ hZ
gh
Y Z ab g
−
= +
Where:
( )
0,1ZN
g >
0
allows for skewness
h >
0
allows for thick tails
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